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### Note_04M.21-27

Course: ECON 445, Fall 2011
School: Texas A&M
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Word Count: 924

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2. 21 Case Two risky assets Following the same procedure as in case 1, we allocate fraction of fund to risky asset 1 and fraction of fund to risky asset 2. The rate of return on portfolio is and its mean and variance are Both the mean and variance are affected by allocation weights and . The relation between and that describes the PPC can be derived as follows. (i) solve the mean equation for to derive...

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2. 21 Case Two risky assets Following the same procedure as in case 1, we allocate fraction of fund to risky asset 1 and fraction of fund to risky asset 2. The rate of return on portfolio is and its mean and variance are Both the mean and variance are affected by allocation weights and . The relation between and that describes the PPC can be derived as follows. (i) solve the mean equation for to derive (ii) Substitute these expressions into the variance equation Though this expression is complicated and is not very useful for our purpose, it shows that the PPC is a hyperbola in and . Two cases need a special attention. The equation for the PPC in the case of a perfectly correlated returns (D=1) becomes a simple equation. Perfect Positive Correlation (D=+1) Perfect Negative Correlation (D=-1). (i) 22 (ii) If , the portfolio has a zero risk ( ). Proof. (A) Perfect Positive Correlation (D=+1). The portfolio possibility curve is a straight line. In this case, the variance equation becomes which gives Solving this for , we can write Substituting this into the equation for the mean return we have And this is a linear equation in the and space, and the PPC is a straight line connecting point A and point B. (B) Perfect Negative Correlation (D=-1). In this case, the variance of the portfolio becomes 23 We need to consider two cases because value can be positive or negative depending on the allocation : (i) positive case: (ii) negative case: Note first that, if , then the portfolio has a zero risk ( . Substituting the solution for from these ) and the mean return is equations into the mean return equations, we derive two segments of the portfolio possibility curves: (i) (ii) QED Numerical method of finding PPC in a general case When the returns are not perfectly correlated, one may find the PPC numerically. Method 1. Recall that, when short sales are not allowed, as one wishes. For each value of , compute must be in a unit interval. Divide the unit interval as finely and by Remark: For the PPC in terms of the mean and variance of final wealth, simply compute Method 2. For a given in the interval from low mean and high mean of the two stocks, compute hyperbola equation presented above: from the 24 Method 3. Excel Solver For a given mean , minimize with respect to subject to the constraint of mean equation. This is not useful at all as we have the simple formula in method 2. Example The figure below shows the portfolio possibility curves of two assets when short sales are not allowed. The mean and variance of the two assets and each curve represents different values of correlation coefficient ranging from -1 to +1. These two assets are taken from one of the examples the in textbook. : F S&P 0.140 0.185 Small Stock 0.17 0.24 Remarks (i) As shown above, the PPC is a straight line when D=+1. The risk of the portfolio cannot be smaller than the risk of either asset. (ii) As D becomes smaller, the PPC bends leftward, and some portfolios can have a smaller risk than the risk of risky assets. How is this possible? (ii) If D=-1, there is a portfolio that has a zero risk ( a proper mix makes it flat. ). As the two assets move in the opposite direction, 25 Minimum Risk Portfolio An extremely risk averse agent may care only about the risk and may choose a portfolio that has the smallest risk. The portfolio of minimum risk is defined by the allocation weight Proof. To find the portfolio of minimum risk, take a derivative of with respect to , set it to zero and solve for The solution for is the fraction given above. Efficient Frontier The portfolio possibility curve above the minimum risk point is called the efficient frontier. The optimum portfolio will be selected only on the efficient frontier. As long as agents prefer a higher expected return and a lower risk, then the portion of the possibility curve below the minimum risk will never be chosen. The efficient frontier is always concave. The efficient frontier cannot have a convex portion because a portfolio consisting of portfolios on the convex portion can dominate. See Chapter 5 for various examples. 26 Optimum Portfolio - Maximum Expected Utility If the objective of the portfolio selection is to minimize the risk (F), then the minimum risk portfolio shown above is the answer. If the utility function is a CARA or a quadratic function, then we need to find the portfolio that maximizes the expected utility. and in the figures below are the expected utility maximizing mean and risk. This is the tangent point of the PPC and one of the indifference curves. Given these optimum values, we need to find the allocation weight that produce the optimum mean and risk. We will consider the expected utility of CARA utility function under the normality. The expected utility is where , We have shown that the maximization of this expected utility is equivalent to the maximization of . The first order condition is From the expressions of mean and variance, we have 27 Substituting these into the first order condition, we can derive There is no guarantee that this solution is in the unit interval. If short sales are not allowed, the weights have to be in the unit interval. Therefore, if this solution is greater than 1, we take 1 as the solution. If this solution is negative, we take zero as the solution. If short sales are allowed, 1) or greater than 1 (short sales of asset 2). can be negative (short sales of asset
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